For the other direction, assume that the class \(\mathcal {H}\)  does not have an infinite multiclass Littlestone tree.
Before we describe the winning strategy of the learner, we need to introduce the notion of ordinal multiclass Littlestone dimension.
We will assign an ordinal to every finite multiclass Littlestone tree. For some preliminaries on ordinals and transfinite recursion, we refer to [1]}. The rank is defined by a partial order \(\prec \) . We set \(t^{\prime } \prec t\)  if \(t^{\prime }\)  is a multiclass Littlestone tree that extends \(t\)  by one level, i.e., \(t\)  is obtained from \(t^{\prime }\)  by removing its leaves. A multiclass Littlestone tree \(t\)  is minimal if it cannot be extended to a multiclass Littlestone tree of larger depth. For such a tree, we set \(\mathrm {rank}(t) = 0\) . If the tree \(t\)  is non-minimal, then it can be extended and this is quantified using transfinite recursion by
\(\mathrm {rank}(t) = \sup \lbrace  \mathrm {rank}(t^{\prime }) + 1 : t^{\prime } \prec t \rbrace \,.\) 
